Post #2704904

#Polyhedron #Polygons #tiles #opticalillusion #MathArt #MathsArt The achromatic axis is the diagonal connecting \((0,0,0,0)\) to \((1,1,1,1)\). The color tesseract has 2 achromatic vertices and 14 chromatic vertices. The saturation of a \((\chi_1,\chi_2,\chi_3,\chi_4)\) color is its distance from the achromatic axis. Maximum saturation occurs at the 14 chromatic vertices Removing the achromatic vertices and their edges from the tesseract leaves behind a nonconvex square dodecahedron. The standard basis of \(\mathbf{R}^4\) are 4 chromatic vertices. They are the primary hues. The 6 sums of pairs of primary hues are the secondary hues. The 4 sums of triples of primary hues are the tertiary hues. This is all 14 chromatic vertices. Each hue achieves its maximum saturation in this dodecahedron. To visualize the square dodecahedron we can project it to the orthogonal complement of the achromatic axis. This turns the square dodecahedron into the Catalan solid known as a rhombic dodecahedron. It is shown in the figure. Right now we are only interested in its surface. A rhombic dodecahedron can be built by attaching the base of 6 square pyramids to the faces of a cube. The triangular faces of adjacent pyramids have to be coplanar so they can join to form the 12 rhombi. The edges of adjacent pyramids are the edges of the cube inscribed in the rhombic dodecahedron. (4/6)